Distance function from x to the Cantor set

1. Sep 18, 2008

Dragonfall

Does the said function satisfy:

(1)continuity
(2)never constant
(3)has uncountably many zeroes

1 and 3 is trivial, but I'm not sure about 2.

2. Sep 22, 2008

Dragonfall

Hello hello

3. Sep 22, 2008

CRGreathouse

I'm not sure how you'd even define the distance, given that every point in (0, 1) has a point in the Cantor set within epsilon for any epsilon > 0.

4. Sep 22, 2008

Moo Of Doom

Not true. The Cantor set is not dense in (0, 1). For example the point 1/2 is at least 1/6 away from any point in the Cantor (middle-thirds) set. In fact, dist({1/2}, CantorSet} = 1/6.

5. Sep 23, 2008

CRGreathouse

Ah... clearly I was thinking of something else. That'll teach me to post late at night!

6. Sep 23, 2008

Doodle Bob

If x=1/2, then the distance from x to the Cantor middle-third set would be 1/6. If x=0, then the distance would be 0. Hence "not constant".

I find the the use of the word "never" strange since it sounds to be like asserting otherwise the function would be constant on, say, the Tuesdays after a new moon, but not constant all other days.

Possibly what you mean is that there are no open sets on which the function is constant.

7. Sep 24, 2008

Dragonfall

By "never" I mean that there is no interval on which it is constant.

This is was a problem I thought up. My intuition was that since if a function is "continuous", and "never constant", each time you hit a zero you must "wave" up and down in order to hit a zero again. So this will make the number of zeros "countable". But the distance function from x to the cantor set seems to be "continuous and never constant" but has uncountably many zeros.